The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth‑grade students. Scores on the test range from 0 to 500.
Demonstrating the ability to use the mean to solve a problem is an example of the skills and knowledge associated with performance at the Basic level. An example of the knowledge and skills associated with the Proficient level is being able to read and interpret a stem‑and‑leaf plot.
In 2019, 147,400
eighth‑graders were in the NAEP sample for the mathematics test. The mean mathematics score was 𝑥⎯⎯⎯=282. We want to estimate the mean score 𝜇 in the population of all eighth‑graders. Consider the NAEP sample as an SRS from a Normal population with standard deviation 𝜎=40.
If we take many samples, the sample mean 𝑥⎯⎯⎯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score 𝜇
in the population. What is the standard deviation of this sampling distribution?
Give your answer to four decimal places.
The standard deviation of the sampling distribution, also known as the standard error, is calculated using the following formula:
Standard Error (SE) = σ / √n
Where σ is the population standard deviation and n is the sample size.
In this case, σ = 40 and n = 147,400.
SE = 40 / √147,400
SE = 40 / 383.917
To four decimal places, the standard deviation of the sampling distribution is:
SE ≈ 0.1041
The standard deviation of the sampling distribution, also known as the standard error of the mean, can be calculated using the formula:
Standard deviation of the sampling distribution = 𝜎/√(n)
where 𝜎 is the standard deviation of the population and n is the sample size.
In this case, 𝜎 = 40 and n = 147,400. Plugging in these values into the formula, we can calculate the standard deviation of the sampling distribution:
Standard deviation of the sampling distribution = 40/√(147400)
Calculating this expression, we get:
Standard deviation of the sampling distribution = 0.1037
Therefore, the standard deviation of the sampling distribution is approximately 0.1037 (rounded to four decimal places).
To find the standard deviation of the sampling distribution, we can use the formula:
Standard deviation of the sampling distribution = 𝜎 / √(n)
Where 𝜎 represents the population standard deviation and n represents the sample size.
In this case, we are given that 𝜎 (population standard deviation) is 40 and the sample size (n) is 147,400.
Plugging in the values to the formula:
Standard deviation of the sampling distribution = 40 / √(147400)
To calculate this, we can use a calculator or spreadsheet software. Rounding the answer to four decimal places, we get:
Standard deviation of the sampling distribution ≈ 0.1037